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giá trị riêng các thông số α và λ được xác định để làm cho f (η) có thời kỳ 2π và làm cho g (ξ ) biến mất tại ξ = R. chức năng phương thức có thể được viết về các chức năng Mathieu như các sản phẩm của ce hình thức (η, q) Ce (ξ, q) cho các phương thức đối xứng về x-trục và se (η, q) se (ξ, q) cho chế độ chống-đối xứng về trục x. | ELLIPTICAL COORDINATE SYSTEM 0.6 0.4 0.2 -0.2 -0.4 -0.6 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 x axis 1 Figure 10.13 Elliptic Coordinate Grid -1 and g e - a - X cosh 200 0 0 e R where the eigenvalue parameters a and A are determined to make f n have period 2n and make g e vanish at e R. The modal functions can be written in terms of Mathieu functions as products of the form ce q Ce e q for modes symmetric about the x-axis and se n q Se e q for modes anti-symmetric about the x-axis. The functions ce and se are periodic Mathieu functions pertaining to the circumferential direction while Ce and Se are modihed Mathieu functions pertaining to the radial direction. The structure of these functions motivates using the following series approximation for the functions for even modes N M ne f ỵỳ k - - wa- g e Ecos R l -1 2 bi. k 1 l 1 2003 by CRC Press LLC The analogous approximations for the modes anti-symmetric about the x-axis are . . N . . . M nf f n 2sin nk ak g f 2 sin R l bl. k i 1 1 Thus the expressions for both cases take the form N M f n fk n ak and g f gi f bi. k 1 l 1 Let us choose a set of collocation points ni i 1 . n and fj j 1 . m. Then substituting the series approximation for f n into the differential equation gives the following over-determined system of equations NN N y fk ni ak 2fk ni ak - Xcos 2ni J2 fk ni ak 0 i 1 . n. k 1 k 1 k 1 Denote F as the matrix having fk ni as the element in row i and column k. Then multiplying the last equation on the left by the generalized inverse of F gives a matrix equation of the form CA aA - XDA 0 where A is a column matrix consisting of the coefficients a k. A similar equation results when the series for g f is substituted into the differential equation for the radial direction. It reduces to EB - aB XGB 0. The parameter a can be eliminated from the last two equations to yield a single eigenvalue equation WE CW X -WG DW where W AB and the tic mark indicates matrix transposition. By addressing the two-dimensional array W
